On the Noetherian dimension of Artinian modules with homogeneous uniserial dimension

ýIn this articleý, ýwe firstý ýshow that non-Noetherian Artinian uniserial modules overý ýcommutative ringsý, ýduo ringsý, ýfinite R-algebras and rightý ýNoetherian rings are 1-atomic exactly like Zp∞ý. ýConsequentlyý, ýwe show that if R is a right duo (orý, ýa rightý ýNoetherian) ringý, ýthen the Noetherian dimension of an Artinianý ýmodule with homogeneous uniserial dimension is less than or equalý ýto 1ý. ýIn particularý, ýif A is a quotient finite dimensionalý ýR-module with homogeneous uniserial dimensioný, ýwhere R is aý ýlocally Noetherian (orý, ýa Noetherian duo) ringý, ýthen n-dim ýA≤ýý1ý. ýWe also show that the Krull dimension of Noetherian modules isý ýbounded by the uniserial dimension of these modulesý. ýMoreoverý, ýwe introduce the concept of qu-uniserial modules and by using thisý ýconceptý, ýwe observe that if A is an Artinian R-moduleý, ýsuch thatý ýany of its submodules is qu-uniserialý, ýwhere R is a right duo (orý, ýa right Noetherian) ringý, ýthen n-dim ýA≤1.